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Studia Geophysica et Geodaetica

, Volume 34, Issue 1, pp 65–77 | Cite as

Non-hydrostatic model of airflow over irregular topography — Theoretical bases

  • Josef Brechler
Article

Summary

This article deals with some problems connected with the formulation of a non-hydrostatic mesoscale model of airflow in the atmosphere. Due to an irregular surface a terrain-following coordinate system is used and the equations of the model are transformed into this system. Sound waves are eliminated by the use of the anelastic approximation. The influence of boundaries is minimized by the use of open boundary conditions at the lateral boundaries of the computational domain and of the absorbing layer beneath the upper boundary.

Keywords

Boundary Condition Atmosphere Coordinate System Computational Domain Theoretical Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols used

B

subscript indicating variables at the lateral boundaries

c

phase speed

cp

specific heat at constant pressure

Dij

components of deformation tensor

Def

rate of deformation

F

r.h.s. of the diagnostic pressure equation

G

jacobian of transformation

g

acceleration of gravity

H

height of the model domain

Hi

isentropic height scale

Hx

x-component of the heat flux vector

Hz

z-component of the heat flux vector

k

numerical constant (=0·21)

KM,KH

coefficient of turbulent diffusion of momentum and heat

n

normal unit vector with respect to the boundary

N

Brunt-Väisälä frequency

P00

atmospheric pressure at the lowest model level

p

instantaneous value of atmospheric pressure

\(\bar p(z)\)

basic part of atmospheric pressure corresponding to a dry adiabatic atmosphere

p′(z)

difference between\(\bar p(z)\) and the actual hydrostatic state of the atmosphere

p″(xj,t)

perturbation part of atmospheric pressure

Pr

turbulent Prandtl number

Rd

gas constant

Ri

Richardson number

t

time

u

velocity vector

u

horizontalx-component of velocity

ui

i-th component of velocity

u0

value ofu at the inflow boundary

w

vertical component of velocity

x

horizontal cartesian coordinate

\(\bar x\)

horizontal transformed coordinate

xi

i-th component of coordinates

y

horizontal cartesian coordinate

\(\bar y\)

horizontal transformed coordinate

z

vertical cartesian coordinate

\(\bar z\)

vertical transformed coordinate

zG

function approximating the shape of the orography

δij

Kronecker symbol

Θ

potential temperature; the same formalism used as

\(\bar \Theta (z)\)

in the case of pressure

Θ′(z)Θ″(xj,t)Θ*

non-dimensional potential temperature

κ

Poisson constant (=Rd/cp)

λz

vertical wave length

ϱ

density; the same formalism used as in the case of

\(\bar \varrho (z)\)

pressure and potential temperature

ϱ′(z)ϱ″(xj,t)τij

components of the turbulent stress tensor

ω

transformed vertical component of velocity

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References

  1. [1]
    P. Bougeault: A Non-Reflective Upper Boundary Condition for Limited-Height Hydrostatic Models. Mon. Wea. Rev., 111 (1983), 420.Google Scholar
  2. [2]
    J. Brechler: Model proudění přes nerovinný zemský povrch s určením parametrů mezní vrstvy atmosféry. Kandidátská disertační práce, MFF UK Praha, 1987.Google Scholar
  3. [3]
    T. L. Clark: A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation. J. Comp. Phys. 24 (1977), 186.Google Scholar
  4. [4]
    H. C. Davies: Limitation of Some Common Lateral Boundary Schemes Used in Regional NWP Models. Mon. Wea. Rev. 111 (1983), 1002.Google Scholar
  5. [5]
    D. R. Durran, J. B. Klemp: The Effects of Moisture on Trapped Mountain Lee Waves. J. Atmos. Sci. 19 (1982), 2490.Google Scholar
  6. [6]
    D. R. Durran, J. B. Klemp: A Compressible Model for the Simulation of Moist Mountain Waves. Mon. Wea. Rev., 111 (1983), 2341.Google Scholar
  7. [7]
    T. Gal-Chen, R. C. J. Somerville: On the Use of a Coordinate Transformation for the Solution of the Navier-Stokes Equations with Topography. J. Comp. Phys., 17 (1975), 209.Google Scholar
  8. [8]
    B. W. Golding, L. M. Leslie, G. A. Mills: Mesoscale Dynamical Models and Practical Weather Prediction. ESA J., 9 (1986), 181.Google Scholar
  9. [9]
    J. B. Klemp, D. K. Lilly: Numerical Simulation of Hydrostatic Mountain Waves. J. Atmos. Sci., 35 (1978), 78.Google Scholar
  10. [10]
    Y. Mahrer, R. A. Pielke: A Numerical Study of the Airflow Over Mountains Using the Two-Dimensional Version of Virginia Mesoscale Model. J. Atmos. Sci., 32 (1975), 2144.Google Scholar
  11. [11]
    M. J. Miller, A. J. Thorpe: Radiation condition for the lateral boundaries of limited area models. Quart. J. Roy. Meteor. Soc., 107 (1981), 615.CrossRefGoogle Scholar
  12. [12]
    Y. Ogura, N. Phillips: Scale Analysis of Deep and Shallov Convection in the Atmosphere. J. Atmos. Sci., 19 (1962), 173.Google Scholar
  13. [13]
    I. Orlanski: A Simple Boundary Condition for Unbounded Hyperbolic Flows. J. Comp. Phys., 21 (1976), 251.Google Scholar
  14. [14]
    R. A. Pielke: Mesoscale Meteorological Modeling. Academic Press, 1984.Google Scholar
  15. [15]
    P. M. Tag, T. E. Rossmond: Accuracy and Energy Conservation in a 3-D Anelastic Model. J. Atmos. Sci., 37 (1980), 2150.Google Scholar
  16. [16]
    M. C. Tapp, P. W. White: A non-hydrostatic mesoscale model. Quart. J. Roy. Meteor. Soc., 102 (1976), 277.CrossRefGoogle Scholar

Copyright information

© Academia, Publishing House of the Czechoslovak Academy of Sciences 1990

Authors and Affiliations

  • Josef Brechler
    • 1
  1. 1.Department of Geophysics and Meteorology Faculty of Math. and Phys.Charles UniversityPrague

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